Transcript Folie 1 - LAS-CAD GmbH | LASCAD

The FEA Code of LASCAD
Konrad Altmann
LAS-CAD GmbH
Heat removal and thermal lensing constitute key
problems for the design of laser cavities for solid-state
lasers (SSL, DPSSL etc.). To compute thermal effects
in laser crystals LASCAD uses a Finite Element code
specifically developed to meet the demands of laser
simulation.
The thermal analysis is carried through in three
steps:
• Determination of heat load distribution,
• Solution of the 3-D differential equations of heat
conduction,
• Solution of the differential equation of structural
deformation.
Differential Equation of Heat Conduction
Differential equations of conduction of heat
 div (T )(T )  Q( x, y, z)
κ coefficient of thermal conductivity
T temperature
Q heat load distribution
T  TD
Dirichlet boundary condition
Surface kept on constant temperature
Boundary condition for fluid cooling
 T T T 
  h f (Ts  T )

,
,
 n n n 
y
z 
 x
hf film coefficient
Ts surface temperature
Differential Equations of Structural Deformation
Strain-stress relation
1 1
( ij )   x ,  y ,  z T  T0   C ( ij )
E
0  div( ij )
σi,j stress tensor
αi coefficient of thermal expansion
E elastic modulus
1  ui u j 
 ij  

2  x j xi 
εi,j strain tensor,
ui displacement
To solve these differential equations, a finite element
discretization is applied on a semi-unstructured grid.
This terminus means that the grid has regular and
equidistant structure inside the crystal which is fiited
irregularily to the boundaries of the body. See for
instance the case of a rod
Semi-unstructured grid in case of a rod
Semi-unstructured meshing has a series
of useful properties:
• The structured grid inside the body allows for
efficient use of the results with optical codes, for
instance easy interpolation,
• Meshing can be carried through automatically,
• The grids can be stretched in x-, y-, and zdirection,
• High accuracy can be achieved by the use of
small mesh size,
• The superconvergence of the gradient inside the
domain leads to an accurate approximation of
stresses.
Computation of Heat Load Distribution
Computation of heat load can be carried
through in two ways:
1) Use of analytical approximations
2) Numerical computation by the use of ray
tracing codes. LASCAD does not have its own
ray tracing code, but has interfaces to the well
known and reliable codes ZEMAX and
TracePro.
For the analytical approximation of the heat load
supergaussian functions are used.
As an example I am discussing the case of an end
pumped rod with a pump beam being focussed from
the left end into the rod.
In this case the absorbed pump power density can
be described as follows

P
x

Q ( x, y , z ) 
exp  2
C x C y wx w y
wx


SGX
SGY
y
2
wy
P incident pump power
α absorption coefficient
z distance from entrance plane
β heat efficiency
Cx, Cy normalization constants
SGX, SGY supergaussian exponents
SG=2 common gaussian, SG ∞ tophat
wx, wy local spot sizes

  z


Local spot sizes wx and wy are given by
wx 
w  ( z  f )  
wy 
w  ( z  f
and
2
0x
2
0y
2
x
x
2

)

y
y

θ divergence angle
f distance from entrance plane
The pump beam can be defined astigmatic, for instance
common gaussian the x direction and tophat in y
direction. Also pumping from both ends is possible.
With the above equations the heat load in end
pumped crystals can be approximated very closely.
Similarly, side pumping of a cylindrical rod can be
described by the use of analytical approximations
as will be shown now.
Side Pumped Rod
Crystal
Diode
Water
Flow Tube
Reflector
In this case the propagation of the pump beam in a
plane perpendicular to the crystal axis is described
by the Gaussian algorithm. It is assumed that the
transformation of the beam traversing the different
cylindrical surfaces can be described by appropriate
matrices. This issue is described in more detail in
Tutorial No.2.
Two important parameters have to adjusted to
get the correct heat load
α absorption coefficient of the pump light
β heat efficiency of the laser material
By the use of the absorption coefficient the attenuation of the pump light can be described by the use
of an exponential law
I ( z)  I ( z  0) exp( z)
The absorption coefficient can determined experimentally by measuring the transmission through a plate of
the laser material.
Numerically the absorption coefficient can be determined by computing the overlap integral of the emission
spectrum of the laser diode and the absorption spectrum
of the laser material
I ( z) 
2
 f ( ) exp(  ( ) z ) d
e
1
Absorption spectrum of 1 atomic % Nd:YAG
Emission spectra of high power laser diode P1202 of
Coherent, Inc. for different values of diode current at
constant temperature 20° C.
The heat efficiency β of the laser material, also
called fractional thermal load, is the relative amount
of the absorbed pump power density which is
converted into heat load. The heat efficiency is
defined by
Pheat

,
Pabs
where Pabs is the absorbed pump power and Pheat is
the generated thermal load.
The heat efficiency β of the laser material depends on
quantummechanical properties of the laser material and
can determined by the following equation
ηp

p
p 
  1   p (1  l )r
 l 
f
l 

pump efficiency (fraction of absorbed pump
photons which contribute to the population of the
upper laser level)
ηr efficiency of spontaneous emission
ηl efficiency of stimulated emission
λp pump wave length
λl
wave length of lasing transition
λf
averaged fluorescence wave length
Neglecting the difference between λl and λf in a rough
approximation the above expression for the heat
efficiency can be written as
p
  1   p l  r (1  l )  .
l
This equation shows that the heat efficiency
mainly is determined by the ratio λp/λl .
For important laser materials values for the heat
efficiency can be found in the literature. For instance,
for Nd:YAG the value 0.3 usually is found and
delivers reliable results. This value has been checked
in cooperation with German universities, and has
been delivering very good agreement with
measurements for the thermal lens in many cases.
Since for Yb:YAG the lower laser level is close to
ground level the heat efficiency is smaller. A value of
0.11 turned out to deliver good agreement with
measurements for the thermal lens.
As mentioned in the paper "introduction and
overview.ppt" measurements carried through by the
Solid-State Lasers and Application Team (ELSA)
Centre Université d'Orsay, France delivered good
agreement with LASCAD simulation.
Numerical Computation of Heat Load Distribution
Analytical approximations for the absorbed pump
power density are not always sufficient. There are
situations, for instance scattering surfaces of the
crystal, where numerical computation by the use of a
ray tracing code if necessary.
For this purpose LASCAD has interfaces to the ray
tracing codes codes ZEMAX and TracePro.
Both programs can compute the absorbed pump
power using a discretization of the crystal
volume into a rectangular voxels. The pump
power absorbed absorbed in each voxel is
written to a 3D data set which can be used as
input for LASCAD which is interpolating the data
with respect to the grid used by the FEA code.
On the LASCAD CD-ROM the following example can be
found for a flashlamp pumped rod analyzed by the use
of ZEMAX.
After 3D interpolation the heat load shown below is
obtained with LASCAD
Another interesting configuration has been analyzed by one
of our customers by the use of TracePro. Here you can see
a crystal rod which is embedded in a block of copper. The
pump light is coming from a diode bar is entering through
this is slot.
Absorbed pump power density computed by the
use of TracePro
Interpolation is LASCAD delivers this plot
Computation of Stress Intensity
Since the individual components of the stress tensor to
not deliver sufficient information concerning fraction
problems, the stress intensity is being computedwhich is
defined by
 I  MAX 1   2 ,  2   3 ,  3  1 
Here  1 ,  2 und  3 are the components of the stress
tensor with respect to the principal axis. The stress
tensor is a useful parameter to control cracking limits.
Stress intensity in an end pumped rod